
// Make newform 169.3.d.b in Magma, downloaded from the LMFDB on 23 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_169_3_d_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_3_d_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [121, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 11, 11];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_169_d();"
function MakeCharacter_169_d()
    N := 169;
    order := 4;
    char_gens := [2];
    v := [3];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_d_Hecke();"
function MakeCharacter_169_d_Hecke( : Kf := false)
    N := 169;
    order := 4;
    char_gens := [2];
    char_values := [[0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, 1, 0, 0], [-3, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 3], [0, 0, 0, 2], [0, 0, 0, 0], [0, 0, 3, 0], [0, 6, 0, 0], [0, 0, 12, 0], [-42, 0, 0, 0], [0, 12, 0, 0], [0, 0, 0, 15], [0, 14, 0, 0], [0, 0, -49, 0], [0, 0, 0, -1], [-24, 0, 0, 0], [0, 0, 0, 16], [30, 0, 0, 0], [0, -12, 0, 0], [0, -7, 0, 0], [0, 0, 0, 12], [54, 0, 0, 0], [0, -14, 0, 0], [0, 0, 0, -50], [0, 6, 0, 0], [0, 0, -144, 0], [0, 0, -26, 0], [114, 0, 0, 0], [0, 51, 0, 0], [-126, 0, 0, 0], [0, 0, 168, 0], [93, 0, 0, 0], [0, 0, 0, 16], [-35, 0, 0, 0], [0, -20, 0, 0], [0, 0, 0, 51], [80, 0, 0, 0], [0, 0, 0, -36], [0, 0, 0, 28], [0, 0, -114, 0], [0, 0, 129, 0], [0, 0, 182, 0], [30, 0, 0, 0], [0, 0, 0, 54], [0, -77, 0, 0], [0, 0, 276, 0], [231, 0, 0, 0], [0, 105, 0, 0], [0, 58, 0, 0], [0, 0, 0, -27], [0, 0, 429, 0], [0, 77, 0, 0], [0, 0, 0, -114], [0, 0, 42, 0], [0, 0, -261, 0], [270, 0, 0, 0], [186, 0, 0, 0], [0, 0, 0, 15], [0, 0, -374, 0], [0, 0, 0, -14], [0, 0, -66, 0], [0, 0, 0, -49], [0, 0, 0, 168], [0, 0, -234, 0], [93, 0, 0, 0], [0, -92, 0, 0], [0, 6, 0, 0], [0, 0, 429, 0], [-165, 0, 0, 0], [0, 0, 0, -63], [0, -194, 0, 0], [0, 0, 0, -92], [-536, 0, 0, 0], [264, 0, 0, 0], [0, -168, 0, 0], [0, -85, 0, 0], [0, 0, -312, 0], [0, 0, 0, -36], [0, 0, 0, -232], [0, -72, 0, 0], [387, 0, 0, 0], [0, -27, 0, 0], [0, 131, 0, 0], [0, 0, 315, 0], [0, 0, 688, 0], [327, 0, 0, 0], [0, 0, 0, 146], [0, -90, 0, 0], [0, 71, 0, 0], [0, 0, 0, 168], [0, 0, -390, 0], [0, 0, 0, 145], [0, -72, 0, 0], [0, 0, -651, 0], [0, 90, 0, 0], [810, 0, 0, 0], [0, 92, 0, 0], [-843, 0, 0, 0], [-250, 0, 0, 0], [0, 0, 0, -183], [301, 0, 0, 0], [0, 0, 0, 145], [0, 0, -1011, 0], [0, 0, -495, 0], [0, 0, 39, 0], [0, 324, 0, 0], [0, 92, 0, 0], [0, 0, 0, -118], [-648, 0, 0, 0], [75, 0, 0, 0], [-100, 0, 0, 0], [0, -12, 0, 0], [0, -280, 0, 0], [0, 0, 0, -222], [0, 0, 0, 315], [0, 0, 822, 0], [0, 84, 0, 0], [0, 0, 948, 0], [348, 0, 0, 0], [-126, 0, 0, 0], [0, 0, 0, -180], [0, 0, 77, 0], [-726, 0, 0, 0], [0, 0, 0, 250], [0, 300, 0, 0], [0, 0, 0, 0], [0, 0, 0, -36], [0, 0, -348, 0], [0, 0, 702, 0], [0, -27, 0, 0], [0, 0, 0, 54], [0, 1, 0, 0], [0, 0, -170, 0], [504, 0, 0, 0], [0, 0, 0, -244], [0, -168, 0, 0], [0, 383, 0, 0], [0, 0, 0, -114], [0, 0, 1134, 0], [465, 0, 0, 0], [0, -456, 0, 0], [0, -155, 0, 0], [0, 0, 640, 0], [0, 0, 0, 350], [0, 0, 480, 0], [0, 0, 0, -140], [0, 0, 0, -105], [0, 0, -1326, 0], [54, 0, 0, 0], [0, 155, 0, 0], [0, -267, 0, 0], [0, 0, -651, 0], [0, 0, -819, 0], [-198, 0, 0, 0], [0, 0, -885, 0], [-882, 0, 0, 0], [1356, 0, 0, 0], [0, -20, 0, 0], [210, 0, 0, 0], [0, 103, 0, 0], [0, 0, 0, 314], [0, 0, -387, 0], [0, -105, 0, 0], [-1647, 0, 0, 0], [0, 352, 0, 0], [0, 0, 0, -79], [-1550, 0, 0, 0], [-1166, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_3_d_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_169_3_d_b(:prec:=4)
    chi := MakeCharacter_169_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_3_d_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_169_3_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![121, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;